Lemma 4

Let A be a ring and m $\geqslant 1$ an integer. One has

(34)

for every $a$ in W(A).

Let $\rho : B \to A$ be a ring homomorphism satisfying the conditions of Lemma 3 of No. 4. Then $W(\rho) : W(B) \to W(A)$ is a surjective homomorphism of rings, and $\Phi_B : W(B) \to B^N$ is an injective homomorphism. It is therefore enough to prove that one has

(35)

whatever $b$ in W(B) and the integers $m \geqslant 1, n \geqslant 0$ may be. Now one has

whence formula (35).

Soit A a ring. For every integer $m \geqslant 0$, denote by $V_m(A)$ the set of Witt vectors $a = (a_n)_{n \in \mathbf{N}}$ such that $a_n = 0$ for $0 \leqslant n < m$. It is the image of the m-th power $V^m$ of the mapping $V_A$. The formulas

(36)

(37) result from prop. 3 of No. 5 by induction on $m$. They imply that $V_m(A)$ is an ideal of W(A).

In what follows, we shall endow $W(A)$ with the topology $\mathscr{T}$ associated with the filtration $(V_m(A))_{m \in \mathbf{Z}}$. Since $V_m(A)$ is an ideal of $W(A)$ for every $m \in \mathbf{Z}$, the topology $\mathscr{T}$ is compatible with the ring structure of $W(A)$ (TG, III, p. 49, example 3). Let $a \in W(A)$; the sets $a + V_m(A)$, where $m$ runs through $\mathbf{N}$, form a fundamental system of neighbourhoods of $a$ for $\mathscr{T}$. Now, it follows from lemma 4 that $a + V_m(A)$ consists of the Witt vectors $b$ such that $a_i = b_i$ for $0 \leq i < m$. Consequently, $\mathscr{T}$ is none other than the product topology on $A^\mathbf{N}$ of the discrete topology on each of the factors, and $W(A)$ is therefore a separated and complete topological ring (TG, II, p. 17, prop. 10 and TG, III, p. 22, prop. 4).

Let us denote by $\tau_A$ (or simply $\tau$) the mapping of $A$ into $W(A)$ which associates with an element $a$ of $A$ $(a, 0, 0, ...)$ . We have $\Phi_n(\tau(a)) = a^{p^n}$ for every $n \in \mathbf{N}$. For every ring homomorphism $\rho : B \to A$, we have $W(\rho) \circ \tau_B = \tau_A \circ \rho$.